Interaction energy between two charged spheres surrounded by electrolyte. The smaller sphere located inside the larger sphere

TL;DR

This paper derives analytical equations for the electric interaction energy between two charged spheres with electrolyte surrounding them, considering various parameters, and extends the model to multiple small spheres within a larger one.

Contribution

It introduces a generalized analytical model for interaction energy involving charged spheres inside electrolytes, extending Newton's Shell Theorem to complex configurations.

Findings

Interaction energy decreases with higher electrolyte ion concentration.

Interaction energy increases with higher temperature.

The model reduces to classical Shell Theorem at zero electrolyte concentration.

Abstract

By using the recently generalized version of Newton Shell Theorem [3] analytical equations are derived to calculate the electric interaction energy between two charged spheres, a small one is located within a larger one, and each sphere is surrounded outside and inside by electrolyte. This electric interaction energy is calculated as a function of the electrolyte ion concentration, temperature, distance between the centers and size of the spheres. At the same distance between the centers of the spheres the absolute value of the interaction energy decreases with increasing electrolyte ion concentration and increases with increasing temperature. At zero electrolyte ion concentration the derived analytical equation transforms into the result of the Shell Theorem. Finally, the analytical equation is generalized to calculate the total electric interaction energy between a large charged sphere and N small charged spheres (located within the large sphere) where the spheres are surrounded by electrolyte.